Eur. Phys. J. B (2015) 88: 77
DOI: 10.1140/epjb/e2015-50832-0
Description of a dissipative quantum spin dynamics with a Landau-Lifshitz/Gilbert like damping and complete derivation of the classical Landau-Lifshitz equation Robert Wieser
Eur. Phys. J. B (2015) 88: 77 DOI: 10.1140/epjb/e2015-50832-0
THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Description of a dissipative quantum spin dynamics with a Landau-Lifshitz/Gilbert like damping and complete derivation of the classical Landau-Lifshitz equation Robert Wiesera Institut f¨ ur Nanostruktur- und Festk¨ orperphysik, Universit¨ at Hamburg, Jungiusstrasse 11, 20355 Hamburg, Germany Received 2 December 2014 / Received in final form 13 February 2015 c EDP Sciences, Societ` Published online 25 March 2015 – a Italiana di Fisica, Springer-Verlag 2015 Abstract. The classical Landau-Lifshitz equation has been derived from quantum mechanics. Starting point is the assumption of a non-Hermitian Hamilton operator to take the energy dissipation into account. The corresponding quantum mechanical spin dynamics along with the time dependent Schr¨ odinger, Liouville and Heisenberg equation has been described and the similarities and differences between classical and quantum mechanical spin dynamics have been discussed. Furthermore, a time dependent Schr¨ odinger equation corresponding to the classical Landau-Lifshitz-Gilbert equation and two ways to include temperature into the quantum mechanical spin dynamics have been proposed.
1 Introduction The increasing need of faster and more powerful computer technology provides more and more new fields in magnetism like the field of magnonics [1] or the field antiferromagnetic spintronics [2–5]. In all these new fields as well as in the established areas like the ferromagnetic spintronics [6–9] the interest is mostly focused in the dynamics. To describe the dynamics of magnetic structures e.g. spin chains on metallic or non-metallic surfaces [10–12] it is need to have an equation of motion. In the most cases this equation is the Landau-Lifshitz (LL) equation [13]. Together with the Maxwell equations [14] this equation is the main equation of Micromagnetism [15], the field which describes the dynamics of nearly all magnetic devices of our daily life. There are more equations describing the dynamics in magnetism like the Bloch equation [16], the Landau-Lifshitz-Bloch equation [17–19] or the Ishimori equation [20]. However, the most important equation is the LL equation. The goal of this publication is to show that the LL equation can be derived from quantum mechanics which gives new insight in the underlying physics and makes it possible to study the quantum-classical transition. Text books [21,22] as well as previous publications [23–25] describe only the derivation of the precessional term. In all cases, the damping term has been added later phenomenological. However, complete derivations using a classical description are known [13,26]. The effect of the quantum mechanical derivation of the LL equation is that it provides an extension of the classical
Supplementary material in the form of one pdf file available from the Journal web page at http://dx.doi.org/10.1140/epjb/e2015-50832-0 a e-mail:
[email protected]
spin dynamics with damping to its quantum mechanical limit: the paper discusses the similarities and difference between the classical and quantum mechanical description of the spin dynamics and shows that the quantum spin dynamics becomes similar to the classical one in the classical limit → 0, and S → ∞. The manuscript is organized as follows. After this introduction, in Section 2, the equation of motion for a single spin will be derived and extended to a multi-spin system (Sect. 3). In Section 4 the results of the previous sections will be proved by numerical calculations and the last Section 5 describes two ways how to include temperature and quantum fluctuations into the computer simulations. The publication ends with a summary (Sect. 6).
2 Equation of motion It is a well-know fact that a non-Hermitian Hamiltonian ˆ = H ˆ − iλΓˆ , with H ˆ and Γˆ Hermitian operators and H + λ ∈ R0 a constant, lead to energy dissipation [27–29]. On the other hand such a Hamiltonian does not conserve the ˆ norm of the wave function |ψ(t) = exp(−iHt)|ψ 0 : ˆ
ˆ
n = ψ(t)|ψ(t) = ψ0 |e−2λΓ t |ψ0 = e−2λΓ t .
(1)
However, the norm can be conserved by replacing Γˆ by Γˆ − Γˆ . In this case the wave function becomes: ˆ
ˆ
ˆ
|ψ(t) = e−iHt e−λΓ t eλΓ t |ψ0 ,
(2)
which is conserved: n = 1. The corresponding Schr¨ odinger equation is given by: d ˆ − iλ Γˆ − Γˆ |ψ(t). (3) i |ψ(t) = H dt
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The energy dissipation itself depends on λ and Γˆ . In the ˆ and therefollowing we assume that Γˆ is equal to H ˆ = H ˆ − iλ(H ˆ − H). ˆ fore, H With this assumption the Schr¨ odinger equation becomes [30]: d ˆ − iλ H ˆ − H ˆ |ψ(t). (4) i |ψ(t) = H dt
We will see that this Schr¨odinger equation can be seen as the quantum mechanical analog of the classical LandauLifshitz equation, where λ is the damping constant. Now, it is quite easy to show that this equation corresponds to the following Liouville (von Neumann) equation [31]: dˆ ρ ˆ − λ ρˆ, ρˆ, H ˆ . = i ρˆ, H (5) dt
We assume that the system is in a pure state therefore the density operator is given by: ρˆ = |ψ(t)ψ(t)|.
(6)
Equation (5) has already the form of the Landau-Lifshitz equation because the commutator [ , ] in the case of spin systems leads to a vector product. So far, we have derived the quantum mechanical Liouville equation. The next step is the Heisenberg equation. Therefore, we concentrate ourselves first on the time ˆ dependence of the expectation value S: d ˆ d ˆ S = Tr ρˆS . (7) dt dt Including the Liouville equation (5) into equation (7), written in the alternative form: dˆ ρ ˆρ , ˆ − λ ρˆ, H ˆ − 2ρˆHˆ (8) = i ρˆ, H dt +
and get immediately the following differential equation: d ˆ ˆ H ˆ ˆ H ˆ −λ ˆ ˆ S, S = −i S, −2 H S . dt + (9) Here, the [ , ]+ is the anticommutator and we have used the fact that the trace is invariant under cyclic permutations. Furthermore: ˆ ρS ˆ = ˆ ˆ Tr ρˆHˆ m|ψψ|H|ψψ| S|ψψ|m m
=
m
ˆ ˆ ψ|H|ψψ| S|ψψ|mm|ψ
ˆ ˆ ˆ S ˆ = ψ|H|ψψ| S|ψ = H ˆ Tr ρˆS ˆ . = Tr ρˆH
(10)
Then, the Heisenberg equation appears after replacing the ˆ by S: ˆ expectation values S dˆ ˆ ˆ ˆ ˆ ˆ ˆ (11) S = −i S, H − λ S, H − 2HS . dt +
Now, the Heisenberg equation (11) as well as the Liouville equation (5) can be used to derive the Landau-Lifshitz equation. Therefore, we have to insert these equations in (7) and to determine the traces. However, we need to know the exact form of the density operator ρˆ. In the case of a Heisenberg spin system with spin quantum number S the density operator can be given by the following multivector expansion [32]: 1 ˆ ˆ 1 ˆ 1 ˆ 1+ Sm Sm + Sml Sˆml + . . . 2S + 1 nS m n2S ml (12) The first term is the Identity matrix which behaves under rotation like a scalar. The next term is the sum over the three spin matrices Sˆx , Sˆy , and Sˆz . These matrices behave under rotation like a vector. The next term is the sum over bivectors 1 ˆ ˆ 1 Sˆml = Sm , Sl − S(S + 1)δml , (13) 2 3 + ρˆ =
m, l ∈ {x, y, z}, and the next term (not shown) is the sum over trivectors Sˆmlk and so on. All higher order terms have the same scheme as the previous ones but with higher order tensors. The prefactors nS and n2S are given by traces nS = Tr(Sˆa Sˆa ) and n2S = Tr(Sˆab Sˆab ) (these values only depend on the spin quantum number S and are independent of the direction a, b ∈ {x, y, z}). The prefactors of the higher expansion terms are similar. This expansion is highly related to the magnetic multipol expansion [33], and the number of terms depends on the spin quantum number S. In the case S = 1/2 the expansion ends with the vector term
1 ˆ ρˆ = 1+ ˆ σm ˆ σm . (14) 2 m Here, σ ˆm are the Pauli matrices. In the case S = 1 the expansion ends with the bivector term 1 ˆ ˆ 1ˆ 1 ˆ ˆ Sm Sm + Sml Sml , + (15) ρˆ = 1 3 2 m 2 ml
and in the case S = 3/2 the expansion ends with the trivector term, and so on. After this small digression let us come back to the derivation of the Landau-Lifshitz equation. We have already noticed that it is necessary to insert the Liouville equation (5) or the Heisenberg equation (11) in equation (7) to derive a complete solvable equation for the ˆ The result is the following equation expectation value S. ˆ (n ∈ {x, y, z}): for the components of S d ˆ ˆ Sˆn . ˆ Sˆn − λTr ρˆ, ρˆ, H ρˆSn = i Tr ρˆ, H Tr dt (16) Inserting ρˆ in (16) and calculating the traces lead to an ˆ which is similar to the Landau-Lifshitz equation for S
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equation if the Heisenberg Hamilton operator can be ˆ = −Beff · S. ˆ written as external field H During the calculation we can ignore the first term of ρˆ ˆ commutates with any operbecause the identity matrix 1 ator and therefore all the commutators with the identity matrix become zero. Furthermore, we can see that only the vector term m Sˆm Sˆm participates to the differenˆ All the higher order expansion terms tial equation of S. (bivector, trivector, . . .) are not contributing. The reason for that is the conservation of the rank k of a tensor under rotation. Now, each term of the multivector expansion has its own rank k. The expansion starts with the Identity matrix with rank k = 0 (scalar). The vector term has rank k = 1, the bivector rank k = 2, the trivector rank k = 3, and so on. The rank of each additional term increases by a factor 1. ˆ shall beNow, we assume that the trajectories of S have classical. This means that the length of the spin is conserved, and the spin only fulfills precession and relaxation. However, any precession as well as relaxation can be described as a rotation of the coordinate system and we have already mentioned that the rank of a tensor is conserved under such a rotation. This also means that the motion of Sˆn is only described by the tensors Sˆn which are responsible for Sˆn : Sˆn = Tr ρˆSˆn 1 1 ˆ Tr(Sˆn ) + = Sm Tr(Sˆm Sˆn ) 2S + 1 nS m =0
+
d ˆ − iλ H ˆ − H ˆ |ψ(t) = H |ψ(t). (22) i 1 + λ2 dt
This equation can be seen as the quantum mechanical analog of the LLG equation: d ˆ 1 ˆ × Beff − λ S ˆ × S ˆ × Beff . S = S 2 2 dt 1+λ 1+λ (23) The classical LLG equation itself appears by replacing the ˆ by the classical spin S. spin expectation value S Alternatively, equation (22) can be derived in the same manner as equation (4) if we include higher order terms in ˆ [36,37]. description the of non-Hermitian Hamiltonian H
3 Single spin vs. multi-spin system
=nS δnm
1 ˆ Sml Tr(Sˆml Sˆn ) + 0 + . . . + 0 . (17) n2S ml
Now, it is known that the LL equation will lead to unphysical results in the limit of a large damping λ 1 [35]. The reason is that the precessional motion is not influenced by the damping. The equation which holds in the limit of a large damping is the LandauLifshitz-Gilbert (LLG) equation. In the case of the LLG equation both terms the precessional term as well as the relaxation term are influenced by the damping. The easiest way to obtain the LLG equation from the LL equation is to make the following transformation of the time t: t → t/(1 + λ2 ). The same transformation in the case of Schr¨ odinger equation (4) leads to:
=0
Eq. (18)
The higher order expansion terms disappear due to (18) Tr Sˆαk Sˆβk = nSk δαβ δkk .
k and k are the rank of the tensor and α and β its components: e.g. Sˆ31 = Sˆz , or Sˆ42 = Sˆzx . ˆ ˆ Inserting the vector term ˆ in equam Sm Sm of ρ tion (16) we find after some algebra: d ˆ ˆ × Beff − λ S ˆ × S ˆ × Beff Sn = S , dt n n (19) or written as complete vector: d ˆ ˆ × Beff − λS ˆ × S ˆ × Beff . (20) S = S dt
This equation is similar to the classical Landau-Lifshitz (LL) equation [34]: d S = S × Beff − λS × (S × Beff ) . dt
(21)
A complete evidence of the previous statement and details of the calculation can be found in the Supplementary of this publication .
This could be the end of the story. However, there is one point which has not been discussed so far. In the classical description we assume a constant length of the spin: |S| = 1. The LL and LLG equations at zero temperature just describe a precessional motion resp. relaxation due to an effective field Beff . In the quantum mechanical descripˆ = ψ|S|ψ ˆ tion we deal with the expectation values S which strongly depend on the wave functions |ψ. The ˆ of these expectation values have not absolute value |S| ˆ ≤ S. Furthermore, due necessarily to be constant: |S| to the time dependence of the wave function |ψ = |ψ(t), ˆ is also time dependent. This means that in general |S| we cannot expect an agreement of the classical with the quantum mechanical spin dynamics. On the other hand, we find in textbooks the statement that we can expect a classical behavior, at least for a single spin in an external field and without damping [22]. This textbook statement of a semiclassical description of the spin dynamics is not totally wrong. In the case of single spins and in the case of linear excitations of spin system in the ferromagnetic ground state (spin wave excitation) the classical and the quantum mechanical description can lead to the same traˆ n (n stands for jectories if the Hamiltonian is linear in S the lattice site and not for a spin component. For details see Ref. [25]). To specify the description let us first assume a single spin which is in a pure state |ψ(t). In this case the density operator is given by equation (6) and we find
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Tr(ˆ ρ2 ) = Tr(ˆ ρ) = 1. For comparison, in the case of a mixed state: ρˆ = pi |ψi ψi |. (24) i
2
We find ρˆ = ρˆ, therefore Tr(ˆ ρ2 ) < Tr(ˆ ρ) = 1. Then, we 2 can write Tr(ˆ ρ ) in the case of a single spin with S = 1/2: Tr(ˆ ρ2 ) =
ˆ 2 1 + |σ| ≤ 1. 2
(25)
A similar formula can be given for higher spin quantum numbers (S > 1/2) due to equation (18). However, in these cases it is impossible to give a geometric plot as for S = 1/2 (Bloch sphere). Tr(ˆ ρ2 ) = 1 appears in the case of a pure state which ˆ = /2. In the case ˆ = 1 resp. |S| corresponds to |σ| of a spin ensemble in a mixed state we have Tr(ˆ ρ2 ) < 1 ˆ corresponding to |S| < /2. This means, in the case of a single spin in a pure state the norm of the expectaˆ is constant and the dynamics can become tion value |S| classical (see Eq. (23)). In the case of a multi-spin system the wave function is described by a product of the eigenstates of the single spins: |ψ = |S1 m1 ⊗ |S2 m2 ⊗ . . . ⊗ |SN mN .
(26)
Such a state is called product state. For example the ferromagnet state is described by a product state: |FM = |↑↑ . . . ↑ = | ↑ ⊗ | ↑ ⊗ . . . ⊗ | ↑. However, product states are normally not the general states of a magnetic system. The normal state in a quantum mechanical system with more than one spin is a superposition of the product states (26): |ψ = cm1 m2 ...mN m1 ...mN
× (|S1 m1 ⊗ |S2 m2 ⊗ . . . ⊗ |SN mN ) .
(27)
The product states in this case cannot be separated which that we cannot write cm1 m2 ...mN = cm1 cm2 . . . cmN . This means that we have a correlation between the spins which do not exist in classical physics. This phenomenon is called entanglement [38,39] and leads to the fact that in the case of two entangled spins a measurement on the first spin also affects the second spin [40]. The same is true for more then two spins. In this case the measurement on one spin affects all spins. Only in the case of a product state the subsystems are not correlated and can be separated, which means that the measurement of a single spin in this case only affect this spin and do not change the wave functions of the other spins as in the case of entanglement. ˆ n , where n is the lattice Now, the expectation value S site, can be calculated similar to a single spin by: ˆ n |ψ, ˆ n = ψ|S S where |ψ is given by (26) or (27), or by: ˆ n = Tr ρˆS ˆn . S
(28)
(29)
In the following we assume a pure state. Therefore, the density operator of our multi-spin system ρˆ is defined by ρˆ = |ψψ|. Alternatively, we can calculate the expectation value ˆ n with aid of the reduced density operator ρˆmn [38]: S ˆn , ˆn ˆ n = Trn Trj=n ρˆS = Trn ρˆmn S (30) S
where Trn is the partial trace over the sub Hilbert space Hn and Trj=n the partial trace over all the other sub Hilbert spaces: H = H1 ⊗ H2 ⊗ . . . ⊗ HN . The advantage of using the reduced density operator ρˆmn is that we have to deal with smaller matrices, which reduces the numerical effort. It is easy to verify that the reduced density operator ρˆmn is given by: ρˆmn = pmn |Sn mn Sn mn |. (31) mn
Equation (31) clearly shows that the reduced density operator corresponds to a mixed state, the pmn are the probabilities to find the system in the state |Sn mn , even if the complete system is in a pure state ρˆ = |ψψ|. In the case of the product state (26) the reduced density operator is given by ρˆmn = |Sn mn Sn mn |. Here, the reduced density operator ρˆmn as well as the density operator of the complete system ρˆ = |ψψ| are pure. ˆ n | = S as long The consequence is that we find: |S as the system is described by a product state. However, as mentioned before the normal states are superposition states which are entangled and not product states. In these ˆ n | < S (see discussion before about the cases we find |S density operators of pure and mixed states). The explicit ˆ n | corresponds to the strength of the entanvalue of |S ˆ n | = S which is glement: no entanglement means |S ˆ n | decreases the maximal value and with entanglement |S and can become zero. Due to the fact that the entanglement depends on |ψ and |ψ can change with the time t, ˆ n | can also change with the time. |S Another way to quantify the entanglement is the von Neumann entropy: S(ˆ ρmn ) = −Tr (ˆ ρmn log2 ρˆmn ) ,
(32)
where ρˆmn is the reduced density matrix. The von Neumann entropy is the extension of the classical Gibbs entropy concept to the quantum mechanics and can be seen as the quantum mechanical analog to the classical Shannon entropy [41] of the information technology. In principle the von Neumann entropy is defined for any density matrix ρ˜. In general, the von Neumann entropy proves if the system is in a pure or in a mixed state. Therefore, it makes sense to use the reduced density matrix ρˆmn because in these cases a pure state corresponds to a product state, and therefore to no entanglement. In these cases we find S(ˆ ρmn ) = 0. A mixed state corresponds to an entangled superposition state. In these cases
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we find 0 < S(ˆ ρmn ) ≤ 1. The maximum value S(ˆ ρmn ) = 1 ˆ n | = 0. Furthermore, as before |S ˆ n |, corresponds to |S S(ˆ ρmn ) is time dependent.
4 Numerical proof
n
0.5
Sˆα
In the previous section we have seen that only in the cases of a single spin or a spin ensemble which behaves like a single spin and in the cases of a linear excitation around the ferromagnetic ground state we can find a quantum mechanical spin dynamics similar to the classical spin dynamics. In these cases we have no entanglement and the ˆ stays constant. In the cases of entanabsolute value |S| glement this is not the case. To proof this statement computer simulations have been performed. In the previous publications only linear excitations [25] or single spins [31] have been investigated. In the following, we will discuss the magnetization reversal of a single spin as well as a linear excitation and a magnetization reversal of a trimer. The Heisenberg Hamiltonian of our system is given by: ˆ = −J ˆ nS ˆ n+1 − μS H S Bz Sˆnz − μS Bx (t)Sˆ1x . (33)
Bx (t) = B0x e
− 12
t−t0 TW
2
.
(34)
This field pulse will be used to bring the system out of equilibrium and to start the reversal process or just for linear excitations. For the moment we restrict ourselves to a single spin (no exchange term in (33)). The spin is initially oriented in +z-direction: |ψ(t = 0) = | ↑. The static external field in −z-direction. The field pulse is needed to break the symmetry and to initialize the magnetization reversal. ˆ calculated with Figure 1 shows the trajectories of S the Schr¨ odinger equation (22) and for comparison the trajectory of the classical spin S calculated with the classical Landau-Lifshitz-Gilbert equation. As expected, Figure 1 shows a perfect agreement of both, the classical as well as quantum mechanical trajectories, which also means ˆ is conserved. For a detailed discussion please see that |S| Sections 2 and 3. Let us come to the trimer. All spins are aligned in a chain and we assume that there is no coupling between the first and the last spin. The Hamiltonian (33) is linear ˆ n and therefore we can expect a classical behavior if in S we are close to the ferromagnetic ground state. Figure 2 shows the trajectories of three ferromagnetic coupled spins, excited by a tiny field pulse (linear excitation), with S = 1. The field pulse acts on the first spin
-1 -0.5 0 0.5 1 Sˆx
0
Sˆx Sˆy Sˆz
-0.5
-1 0
10
5
Sx Sy Sz
15
time t
Fig. 1. Magnetization reversal of a single spin. For comparison the quantum mechanical expectation values Sˆn n ∈ {x, y, z} and the classical spin components have been plotted as function of time. The inset shows the precessional motion during the reversal. (S = 1, J = 0, μS Bz = −5.1, μS B0x = 3.27, t0 = 2, TW = 0.02, λ = 0.2). 0.2 0.15
n
The first term describes the nearest neighbor exchange interaction with ferromagnetic (J > 0) or antiferromagnetic (J < 0) coupling. The second sum and third term describe couplings to external magnetic fields. In this case a static field in z-direction and a field pulse in x-direction acting on the first spin only:
1 0.5 Sˆy 0 -0.5
1
ˆ1 S ˆ2 S ˆ3 S
S1
0.1 Sˆny
0.05 0 -0.05 -0.1
-0.05
0
0.05
0.1
Sˆnx
Fig. 2. Linear excitation and additional relaxation of three ferromagnetic coupled spins: only the first spin has been excited by the field pulse. A perfect agreement between quantum mechanical and classical trajectory can be seen for the first spin. The same is true also for the other spins. (S = 1, J = 1, μS Bz = 0.1, μS B0x = 3.27, t0 = 10, TW = 0.02, λ = 0.1).
only. The initial configuration is: all spins are oriented in +z-direction, the direction of the external field Bz . To show the agreement of the classical spins Sn with ˆ n we have compared the curves the expectation values S of the first spin: it can be seen that both curves lie on top of each other which means that the expectation values behave classical. The comparison between classical and quantum mechanical trajectory of the other two spins shows the same behavior (not shown). As mentioned before such an agreement can be expected only for linear excitations of the ferromagnetic ground state, in this case |FM = |↑↑↑. This is the case within the simulation. This can be seen by the small amplitude of the Sˆnx and Sˆny component. Furthermore, the simulation shows that only the basis state |Sm1 ⊗ |Sm2 ⊗ |Sm3 = |m1 m2 m3 = |↑↑↑ gives
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1
1
Bx-pulse
Bx-pulse
m1m2m |ψ(t)
0.8
0.5
↑↑↑ |ψ(t)
↓↓↓ |ψ(t)
0.6
Sˆnz
0
0.4 + 1 |ψ(t) 2
-0.5
0.2
1 -1 0
c)
− 12 |ψ(t)
2 20
3
0
40
60
time t
80
0
100
20
40
60
time t
80
100
d)
1
0.75
1
0.5
0.8
entropy S(ρˆm1 )
a)
Eur. Phys. J. B (2015) 88: 77
0.8 0.6 0.4 0.2
0.25
Sˆ2α
0
ˆn S
-0.25 -0.5
Sˆ2x Sˆy
-0.75 -1 0
0.6
0 0
0.4 0.2
20
2
20
40
60
time t
80
100
0 0
20
40
60
time t
80
100
Fig. 3. Magnetization reversal for three ferromagnetic couple spins. (a) z-component of the spin as function of time t, (b) projection of the wave function |ψ(t) to the basis states |m1 m2 m3 , (c) precession motion of the second spin during the reversal process, (d) spin length as function of time t (S = 1/2, J = 4, μS Bz = −2, μS B0x = 3.27, t0 = 10, TW = 0.02, λ = 0.1).
a relevant impact: |↑↑↑|ψ(t)|2 ≈ 1. All other basis states |m1 m2 m3 are not occupied or just marginal with |m1 m2 m3 |ψ(t)|2 < 0.01. Furthermore, additional simulations show that with increasing S the deviation from |ψ(t) = |FM, up to which we can see a classical behavior, increases. This becomes clear because with S → ∞ we find the classical limit. The situation changes if we leave the ferromagnetic ground state. Figure 3 shows the magnetization reversal of three coupled spins with S = 1/2. The initial configuration is the same as before all three spins in +z-direction, however the external field is now −z-direction. Again, the field pulse excites the first spin only. The result is a stepwise magnetization reversal (see Fig. 3a). Each step corresponds to the occupation of one of the basis states |m1 m2 m3 corresponding to | + 3/2 = |↑↑↑ (basis state with three spins up), | + 1/2 (all basis states with two spins up), | − 1/2 (all basis states with two spins down), and | − 3/2 = |↓↓↓ (basis state with three spins down) (see Fig. 3b). The oscillation of Sˆ1z and Sˆ3z around Sˆ2z is a direct result of the field pulse acting on the first spin only and the exposed position of the second spin as the middle of this three spin cluster. During the reversal process Sˆnx and Sˆny (Fig. 3c) show a precessional motion of the spins similar to the magnetization reversal of the single spin before (see Fig. 1). The most important result is shown in Figure 3d: the ˆ n |. This means, we cannot not conserved absolut value |S expect an agreement with the classical trajectories where we assume a constant spin length |Sn | = 1. As mentioned ˆ n | < S means that the system is entangled, before |S which can be seen also by Figure 3b. The three spins are in a product state (FM) only at the beginning and at the end of the reversal process. In between, we see superpositions
40
60
time t
80
100
Fig. 4. Von Neumann entropy S(ˆ ρm1 ) vs. time t, corresponding to the magnetization reversal of the three spin system presented in Figure 3.
of the basis states |m1 m2 m3 = |Sm1 ⊗ |Sm2 ⊗ |Sm3 which means entanglement. To strengthen this message the von Neumann entropy S (ˆ ρm1 ) = −Tr (ˆ ρm1 log2 ρˆm1 ) ,
(35)
for the reduced density operator ρˆm1 , corresponding with the Hilbert space of the first spin has been calculated. For the details of the calculation please see the Supplementary material . Figure 4 shows the von Neumann entropy S(ˆ ρm1 ) as function of time t. It can be seen that ferromagnetic states at the beginning and end: |↑↑↑ and |↓↓↓ show no entanglement. Furthermore, we see that the highest entangleˆ n |, n ∈ {1, 2, 3}, ment [max. S(ˆ ρm1 )] appears when the |S have their smallest values.
5 Temperature The previous sections describe the quantum spin dynamics with energy dissipation at zero temperature. Within this section we discuss two possibilities to include temperature effects. The easiest way is to add to the Hamilton ˆ (Eq. (33)) an additional stochastic field term: operator H ˆξ = − ˆ n. H ξn S (36) n
ξn = (ξnx , ξny , ξnz ) is a stochastic field characterized by β (t ) = Dδnm δαβ δ(t − t ), with ξnα (t) = 0, and ξnα (t)ξm α, β ∈ {x, y, z}, and lattice sites n, m. Depending on the prefactor D the stochastic field can be used to describe temperature or quantum fluctuations [42]. The advantage of such a stochastic field is that this term is already a field term. This means that this term behaves classical. In other words: the Heisenberg ˆ = −(Beff + ξ)S ˆ inserted in the Hamiltonian H Schr¨ odinger equation (22) immediately leads to the following Langevin equation which can be seen as the quantum
Eur. Phys. J. B (2015) 88: 77
Page 7 of 8
mechanical analog of the stochastic Landau-LifshitzGilbert equation [43,44]: d ˆ 1 ˆ × Beff − λ S ˆ × S ˆ × Beff S S = dt 1 + λ2 1 + λ2 1 ˆ × ξ− λ S ˆ × S ˆ × ξ . (37) + S 1 + λ2 1 + λ2
Similar, the quantum mechanical analog of the stochastic Landau-Lifshitz can be derived using the same Heisenberg Hamiltonian together with the Schr¨ odinger equation (4) or by making the same transformation we have used to derive the Schr¨ odinger equation (22). An alternative way to include temperature, where we assume that the system is already in equilibrium, is to use statistical operator [38,45]: ˆ
ρˆStat. =
e−β H , ˆ Tr e−β H
with β the well-known inverse temperature β = Then, the time dependence of ρˆStat. is given by: ˆ (t)ˆ ˆ + (t). ρˆStat. (t) = U ρStat. (0)U
(38) 1 kB T
.
(39)
ˆ (t) is the unitary operator U ˆ ˆ ˆ ˆ U(t) = e−iHt e−λHt eλHt ,
(40)
which we have already seen in equation (2). At this point, we have to notice that equation (39) is a self-consistent ˆ (t) contains H ˆ equation, because the unitary operator U ˆ ˆ ρStat. H). which has to be calculated with ρˆStat. : H = Tr(ˆ ˆ to be calculated. AlterHowever, ρˆStat. already needs H natively, we can use (38) in the Liouville equation (5) to describe the dynamics.
It has been shown that only for single spins with a ˆ and in the case of a ˆ = −B · S Zeeman Hamiltonian H ferromagnetic multi-spin system we can expect a classical behavior, which means similar trajectories of the quantum ˆ n and classical spins Sn . mechanical expectation values S (The index n stands for the nth spin.) In all other cases we expect a deviation of the trajectories of the expectation ˆ n from the classical behavior. The reasons are values S the appearance of additional terms of the order of in the Heisenberg equation which disappear in the classical limit: S → ∞ and → 0 and in the case of a multi-spin system the appearance of entanglement. The entanglement itself is a pure quantum effect and also disappears in the classical limit. All the results have been discussed theoretically with analytical calculations and proved with computer simulations. In all cases the theoretical estimated effects have been seen in the simulations. In the last section of the manuscript two ways have been shown how to include temperature which has not been taken into account in the previous sections (Sects. 2–4). The first way is to add a random noise. This way allows us to investigate the dynamics under the influence of temperature as well as quantum fluctuations. The second way deals with the statistical operator. In this case fluctuations play no role. This way can be used if we are interested in systems which are always in the thermodynamical equilibrium. This means, non-equilibrium effects cannot be described with this way. The author thanks D. Altwein, M. Krizanac, E. Vedmedenko, Y. Saadi, M. Maamache, and P.-A. Hervieux for helpful discussions and comments. This work has been supported by the Deutsche Forschungsgemeinschaft in the framework of subproject B3 of the SFB 668 and by the Cluster of Excellence “Nanospintronics”.
References 6 Summary In summary it has been show that the following nonˆ = H−iλ ˆ ˆ inserted in the Hermitian Hamilton operator H H time dependent Schr¨odinger equation leads to an equation of motion similar to the classical Landau-Lifshitz equation (21). To show this the corresponding Liouville equation (5) as well as Heisenberg equation (11) have been derived and discussed. It is known that the Landau-Lifshitz equation fails in the huge damping limit [35]. However, the LandauLifshitz-Gilbert (LLG) equation can be simply obtained from the Landau-Lifshitz equation by rescaling the time. The LLG equation itself shows the correct physics. The same rescaling has been used to derive the time dependent Schr¨ odinger equation (22) which corresponds to the classical Landau-Lifshitz-Gilbert equation (23). During the complete derivation and later in the manuscript the similarities and differences of the classical and quantum spin dynamics have been discussed.
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Page 8 of 8 12. M. Menzel, Y. Mokrousov, R. Wieser, J.E. Bickel, E. Vedmedenko, S. Bl¨ ugel, S. Heinze, K. von Bergmann, A. Kubetzka, R. Wiesendanger, Phys. Rev. Lett. 108, 197204 (2012) 13. T.L. Gilbert, IEEE Trans. Mag. 40, 3443 (2004) 14. J.D. Jackson, Classical Electrodynamics (John Wiley & Son, New York, 1999) 15. W.F. Brown, Micromagnetics (Wiley, New York, 1963) 16. F. Bloch, Phys. Rev. 70, 460 (1946) 17. D. A. Garanin, Physica A 172, 470 (1991) 18. D. A. Garanin, Phys. Rev. B 55, 3050 (1997) 19. R.F.L. Evans, D. Hinzke, U. Atxitia, U. Nowak, R.W. Chantrell, O. Chubykalo-Fesenko, Phys. Rev. B 85, 014433 (2012) 20. Y. Ishimori, Prog. Theor. Phys. 72, 33 (1984) 21. H.C. Fogedby, Theoretical Aspects of Mainly Low Dimensional Magnetic Systems (Springer-Verlag, Berlin, Heidelberg, 1980) 22. B. Guo, S. Ding, Landau-Lifshitz Equations (World Scientific, Singapore, 2008) 23. M. Lakshmanan, Phil. Trans. R. Soc. A 369, 1280 (2011) 24. A. Sakuma, arXiv:cond-mat/0602075v2 (2006) 25. R. Wieser, Phys. Rev. B 84, 054411 (2011) 26. M. F¨ ahnle, D. Steiauf, J. Seib, J. Phys. D 41, 164014 (2008) 27. V. Weisskopf, E. Wigner, Z. Phys. 65, 18 (1930) 28. R.L. Liboff, M.A. Porter, Physica D 195, 398 (2004) 29. R. Kosik, Ph.D. thesis, TU Wien, 2004
Eur. Phys. J. B (2015) 88: 77 30. N. Gisin, Helv. Phys. Acta 54, 457 (1981) 31. R. Wieser, Phys. Rev. Lett. 110, 147201 (2013) 32. H.F. Hofmann, S. Takeuchi, Phys. Rev. A 69, 042108 (2004) 33. E.Y. Vedmedenko, N. Mikuszeit, ChemPhysChem 9, 1222 (2008) 34. D.L. Landau, Phys. Z. Sowjetunion 2, 46 (1932) 35. R. Kikuchi, J. Appl. Phys. 27, 1352 (1956) 36. D. Altwein, Ph.D. thesis, University of Hamburg, 2015 37. R. Wieser, Domain wall dynamics in quasi onedimensional nanostructures (S¨ udwestdeutscher Verlag f¨ ur Hochschulschriften, Saarbr¨ ucken, 2014) 38. D.A. Garanin, Adv. Chem. Phys. 147, 213 (2011) 39. P. Krammer, Ph.D. thesis, Universit¨ at Wien, 2009 40. A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935) 41. C.E. Shannon, The Bell System Technical Journal 27, 379, 623 (1948) 42. V.L. Pokrovsky, N.A. Sinitsyn, Phys. Rev. B 67, 144303 (2003) 43. J.L. Garc´ıa-Palacios, F.J. L´ azaro, Phys. Rev. B 58, 14937 (1998) 44. U. Nowak, in Annual Reviews of Computational Physics IX, edited by D. Stauffer (World Scientific, Singapore, 2001), p. 105 45. K. Blum, Density Matrix Theory and Applications (Plenum Press, New York, London, 1996)
Supplement: Quantum spin dynamics R. Wieser Institut f¨ ur Angewandte Physik und Zentrum f¨ ur Mikrostrukturforschung, Universit¨ at Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany (Dated: March 1, 2015) PACS numbers: 75.78.-n, 75.10.Jm, 75.10.Hk
I.
DERIVATION OF THE LANDAU-LIFSHITZ EQUATION FOR S = 1
P To prove the given statement that only the vector term m hSˆm iSˆm of ρˆ will contribute to the description of the dynamics of hSˆn i, n ∈ {x, y, z}, we have to determine the following equation of motion: d ˆ ˆ Sˆn , ˆ Sˆn − λTr [ˆ (1) ρ, [ˆ ρ, H]] Tr ρ, H] ρˆSn = iTr [ˆ dt for a certain S and taking into account all expansion terms of ρˆ: 1 X ˆ 1 X ˆ ˆ 1 ˆ hSm iSm + hSml iSˆml + . . . , 1+ ρˆ = 2S + 1 nS m n2S
(2)
ml
with m, l ∈ {x, y, z}. In the following we set S = 1, which means ρˆ is given by: ρˆ =
1ˆ 1 X ˆ ˆ 1X ˆ 1+ hSm iSm + hSml iSˆml . 3 2 m 2
(3)
ml
P σij iˆ σij is already An extension to S > 1 will lead to the same results. In the case of S = 1/2 the bivector term ij hˆ zero due to the Clifford algebra, which tells us that σ ˆi σ ˆj = −ˆ σj σ ˆi and therefore [ˆ σi , σ ˆj ]+ = 0. Inserting (3) in (1) gives for the left hand side of (1): 1 X dhSˆ i 1 X dhSˆ i ˆ d ˆ 1 d1 m ml Tr (4) ρˆSn = Tr Sˆn + Tr Sˆm Sˆn + Tr Sˆml Sˆn . dt 3 dt | {z } 2 m dt | dt | {z } 2 ml {z } =0
With aid of the rule:
=0
=2δnm
′ Tr Sˆαk Sˆβk = nSk δαβ δkk′ ,
(5)
and the fact that the spin matrices Sˆn are traceless: Tr Sˆn = 0 we find for the left hand side of (1) the expected result: d ˆ d Tr ρˆSn = hSˆn i . (6) dt dt ˆ Sˆn on the right hand side of (1) becomes: After inserting (3) in (1) the first term iTr [ˆ ρ, H] iX X X a a a ˆ Sˆn = − i ˆ Sˆa ]Sˆn − i iTr [ˆ ρ, H] . Beff Tr [1, hSˆm iBeff Tr [Sˆm , Sˆa ]Sˆn − hSˆml iBeff Tr [Sˆml , Sˆa ]Sˆn (7) 3 a 2 ma 2 mla
ˆ is given by: Here, we have assumed that H ˆ =− H
X
a ˆ Beff Sa .
(8)
a
ˆ Sˆa ] = 0. The trace of the second term becomes: The first term on the right hand side of (7) is zero because of [1, (9) Tr [Sˆm , Sˆa ]Sˆn = iTr Sˆb Sˆn ǫmab = 2iδbn ǫmab ,
2 and therefore: −
X iX ˆ a a ˆ × Beff . hSˆm iBeff ǫman = hSi hSm iBeff Tr [Sˆm , Sˆa ]Sˆn = 2 ma n ma
The commutator in the third term can be written as: [Sˆml , Sˆa ] = i Sˆmc ǫlac + Sˆbl ǫmab ,
(10)
(11)
and therefore, we get two terms with Tr Sˆn Sˆml = 0. This means that the trace of the third term becomes zero. In ˆ Sˆn becomes: summary, only the vector term contributes to the description and iTr [ˆ ρ, H] ˆ × Beff ˆ Sˆn = hSi . iTr [ˆ ρ, H] n
(12)
This result has been mentioned in the publication and can be found in [1, 2]. ˆ ˆ ˆ The second term −λTr [ˆ ρ, [ˆ ρ, H]]Sn on the right hand side of (1) is more tricky. Inserting ρˆ [Eq. (3)] and H
[Eq. (8)] we get: X a ˆ Sˆn = − λ −λTr [ˆ ρ, [ˆ ρ, H]] hSˆm ihSˆb iBeff Tr [Sˆm , [Sˆb , Sˆa ]]Sˆn − 4 mba λ X ˆ ˆ a − hSb ihSml iBeff Tr [Sˆb , [Sˆml , Sˆa ]]Sˆn − 4 bmla
λ X ˆ a hSml ihSˆb iBeff Tr [Sˆml , [Sˆb , Sˆa ]]Sˆn (13) 4 mlba λ X ˆ a hSml ihSˆuv iBeff Tr [Sˆml , [Sˆuv , Sˆa ]]Sˆn 4 mluva
ˆ Sˆa ] = 0. Here, we have immediately ignored the identity matrix. These terms will not contribute because of [1, ˆ ˆ ˆ ]] Let our investigation start with the second and third term. In both cases the double commutator ([Sml , [Sb , Sa and [Sˆb , [Sˆml , Sˆa ]]) will end up with four terms containing a bivector Sˆab . Therefore, the traces Tr [Sˆml , [Sˆb , Sˆa ]]Sˆn as well as Tr [Sˆb , [Sˆml , Sˆa ]]Sˆn become zero because Tr Sˆab Sˆn = 0 [see Eq. (5)].
The double commutator [Sˆml , [Sˆuv , Sˆa ]] in the last (fourth) term of Eq. (13) leads to two commutators of the type 1 ˆ ˆ ˆ ˆ [Sˆml , Sˆop ] = (14) [Sm Sl , So Sp ] + [Sˆm Sˆl , Sˆp Sˆo ] + [Sˆl Sˆm , Sˆo Sˆp ] + [Sˆl Sˆm , Sˆp Sˆo ] . 2
Now, every commutator of these four commutators can be written as e.g.:
(15) [Sˆm Sˆl , Sˆo Sˆp ] = Sˆm Sˆo [Sˆl , Sˆp ] + Sˆm [Sˆl , Sˆo ]Sˆp + Sˆo [Sˆm , Sˆp ]Sˆl + [Sˆm , Sˆo ]Sˆl Sˆp . P P The important point is that we have now two identical sums ml and uv . Therefore, we will always find commutator pairs like [Sˆa , Sˆb ] and [Sˆb , Sˆa ] = −[Sˆa , Sˆb ]. This means at the end, we have pairs of identical traces: Tr Sˆa Sˆb Sˆc Sˆn and −Tr Sˆa Sˆb Sˆc Sˆn , just with opposite signs, which neglect each other. Therefore, the sum over all traces gives zero and the last term of Eq. (13) does not contribute to the dynamics of hSˆn i. Let us come to the missing first term of Eq. (13). The traces of this term can be written as: (16) Tr [Sˆm , [Sˆb , Sˆa ]]Sˆn = Tr Sˆm Sˆb Sˆa Sˆn − Tr Sˆm Sˆa Sˆb Sˆn − Tr Sˆb Sˆa Sˆm Sˆn + Tr Sˆa Sˆb Sˆm Sˆn .
From quantum mechanics we known that Tr Sˆa Sˆb Sˆc Sˆd = nS (δab δcd + δad δbc ), and nS = 2 in the case S = 1. Therefore, Tr [Sˆm , [Sˆb , Sˆa ]]Sˆn becomes: Tr [Sˆm , [Sˆb , Sˆa ]]Sˆn = 4 (δan δbm − δam δbn ) = 4ǫrab ǫrmn .
Furthermore: X λX ˆ a a ˆ × hSi ˆ × Beff − . hSˆm ihSˆb iBeff ǫabr ǫrmn = −λ hSi hSm ihSˆb iBeff Tr [Sˆm , [Sˆb , Sˆa ]]Sˆn = −λ 4 n mba
mba
(17)
(18)
3 ˆ Sˆn In summary, again, only the vector term of ρˆ contributes to the equation of motion for hSˆn i and −λTr [ˆ ρ, [ˆ ρ, H]] becomes: ˆ × hSi ˆ × Beff ˆ Sˆn = −λ hSi . (19) −λTr [ˆ ρ, [ˆ ρ, H]] n
This means in total, Eq. (1) becomes:
d ˆ ˆ × hSi ˆ × Beff ˆ × Beff − λ hSi , hSn i = hSi dt n n
(20)
d ˆ ˆ × Beff − λhSi ˆ × hSi ˆ × Beff . hSi = hSi dt
(21)
or written as complete vector equation:
II.
ALTERNATIVE DERIVATION
In the previous section Eq. (21) has derived by inserting ρˆ [Eq. (2)] in Eq. (1). We have seen that only the vector term of ρˆ contributes to the equation of motion. Therefore, we can immediately make the ansatz: ˆ = Px Sˆx + Py Sˆy + Pz Sˆz , ρˆ ≈ P · S
(22)
with P the normalized polarization P=
ˆ hSi . ~S
(23)
ˆ in the Liouville equation: ˆ = −Beff · S/~ Inserting ρˆ and H i dˆ ρ ˆ − λ [ˆ ˆ , = [ˆ ρ, H] ρ, [ˆ ρ, H]] dt ~ ~
(24)
leads immediately to: dPy ˆ dPz ˆ 1 1 1 dPx ˆ Sx + Sy + Sz = (P × Beff )x Sˆx + (P × Beff )y Sˆy + (P × Beff )z Sˆz dt dt dt ~ ~ ~ −λ (P × (P × Beff ))x Sˆx − λ (P × (P × Beff ))y Sˆy − λ (P × (P × Beff ))z Sˆz . (25) It is easy to see that this is nothing else as the vector differential equation for the polarization P spanned with the basis vectors Sˆz , Sˆz , and Sˆz : dP 1 = P × Beff − λP × (P × Beff ) . dt ~
(26)
Here in this section, ~ has been explicitly taken into account and not set ~ = 1 as before. This gives us the possibility ˆ has the dimension of ~: Js. The Beff to discuss the dimensions. The polarization P is dimensionless, because S ˆ ˆ has the dimension J due to H = −Beff · S/~. Therefore, both sides of Eq. (26) have the dimension 1/s if λ has the dimension 1/Js. This is equal to the Landau-Lifshitz equation. Moreover, 1/~ has the same dimension as γ/µS , with γ the gyromagnetic ratio and µS the magnetic moment. III.
SOME REMARKS ABOUT THE COMMUTATOR AND DOUBLE COMMUTATOR
From classical physics we know the general definition of the Poisson bracket: {F, G} =
∂F ∂G Jµν , ∂xµ ∂xν
(27)
with the arbitrary functions F = F (xµ , xν ) and G = G(xµ , xν ) and the symplectic matrix Jµν . Here, we have already assumed the summation convention, which tells us when an index variable appears twice in a single term it implies summation over all the values of the index of that term.
4 As a logical consequence of the definition of the Poisson bracket the double Poisson bracket is defined as: ∂G ∂H ∂F ∂ Jαβ Jµν . {F, {G, H}} = ∂xµ ∂xν ∂xα ∂xβ
(28)
In the case of a spin system xµ has to be replaced by Sn , and the symplectic matrix Jµν is given by Sl ǫnml . Therefore, the Poisson bracket becomes: {F , G} =
∂F ∂G Sl ǫnml , ∂Sn ∂Sm
with F = F (S) and G = G(S) are arbitrary spin functions. The corresponding double Poisson bracket is given by: ∂G ∂H ∂F ∂ Sc Sl ǫabc ǫnml . {F , {G, H}} = ∂Sn ∂Sm ∂Sa ∂Sb
(29)
(30)
The connection between Poisson bracket { , } and commutator [ , ] is given by the transformation [ , ] ↔ i~{ , }. In the case of the double Poisson bracket: [ , [ , ]] ↔ −~2 { , { , }}. ˆ and double commutator [ˆ ˆ of the Liouville equation (24) Then, we can easily calculate the commutator [ˆ ρ, H] ρ, [ˆ ρ, H] u ˆ ˆ using the definitions for ρˆ [Eq. (22)] and H = −Beff Su /~: ˆ ˆ i v ˆ v ∂ Su ∂ Sv ˆ ˆ = 1 Pu Beff Sl ǫnml = Pu Beff Sl δun δvm ǫnml = (P × Beff )l Sˆl , (31) [ˆ ρ, H] ~ ~ ∂ Sˆn ∂ Sˆm ˆ ˆ ˆ ˆ λ b ˆ w ∂ Su ∂ Sv ∂ Sw ∂ Sc ˆ ˆ = −λPu Pv Beff − [ˆ Sl ǫabc ǫnml = −λPn Pa Beff Sl ǫabm ǫnml = −λ (P × (P × Beff ))l Sˆl . ρ, [ˆ ρ, H]] ˆ ˆ ˆ ˆ ~ ∂ Sn ∂ Sa ∂ Sb ∂ Sm (32) IV.
REDUCED DENSITY OPERATOR
The reduced density operator was introduced by Dirac in 1930 [3]. While the normal density operator ρˆ contains the information of the full system H = HA ⊗ HB the reduced density operator ρˆA only the information about the subsystem HA . Here, we have assumed that the complete Hilbert space H can be divided into two subsystems HA ˆ n and the environment (other spins j 6= n or the lattice). This reduction of information and HB , e.g. a single spin S can be an advantage in the description especially for huge systems. The expectation value of an operator Aˆ is given by: ˆ ˆ = Trab ρˆ(ab)A(a) . (33) hAi
ρˆ(ab) is the density operator of the complete system and the trace is explicitly expressed as being evaluated over the ˆ two sets of variables a and b belonging to the two sub systems HA and HB . The operator A(a) is independent of b and the trace is just a diagonal sum. Therefore, (33) can be written as: ˆ ˆ ˆ = Tra Trb [ˆ , (34) ρ(ab)] A(a) = Tra ρˆA (a)A(a) hAi
with the reduced density operator:
ρˆA (a) = Trb [ˆ ρ(ab)] .
(35)
The reduced density operator ρˆA can be used similar to the normal density operator ρˆ, however ρˆA does not contain the full information of the system. To clarify this point it is favorable to investigate the connection between the full and reduced density operator. The wave function of our system can be expanded by product states constructed by the orthonormal eigenvectors of the subsystems: X cij (|ai i ⊗ |bj i) . (36) |ψi = ij
5 Then, the density operator ρˆ(ab) is given by: ρˆ(ab) = |ψihψ| =
X
cij c⋆i′ j ′ (|ai ihai′ | ⊗ |bj ihbj ′ |) .
iji′ j ′
The reduced density operator ρˆA (a) can be evaluated by the trace over the b variables: XX X ρˆA (a) = Trb [ˆ ρ(ab)] = cij c⋆i′ j ′ (|ai ihai | ⊗ hbj ′′ |bj ihbj ′ |bj ′′ i) = cij c⋆i′ j |ai ihai′ | . j ′′ iji′ j ′
(37)
(38)
iji′
Here, we have used the orthonormality of the eigenfunctions |bj i: hbj ′′ |bj i = δjj ′′ and hbj ′ |bj ′′ i = δj ′ j ′′ , further the fact that: M ⊗ s = sM, where M = |ai ihai′ | is a matrix and s = δjj ′′ δj ′ j ′′ = 1 resp. 0 a scalar. For the following discussion it is favorable to make a unitary transformation ρ˜A (a) = U + ρˆA (a)U to make ρˆA (a) diagonal in |ai i [5]. Such a transformation is always possible and does not change the physics, we just change the underlying coordinate system. With such an unitary transformation, we get: X |cij |2 |ai ihai | . (39) ρ˜A (a) = ij
|cij |2 is the probability to find the system in state |ai i and at the same time in state |bj i. The summation over all j: X |cij |2 (40) pi = j
gives the overall probability that the system is in state |ai i regardless of |bj i. Therefore, ρ˜A (a) acts like a normal density operator of a mixed system: X pi |ai ihai | . (41) ρ˜A (a) = i
V.
SOME REMARKS ABOUT PURE AND MIXED DENSITY OPERATORS
In the case of a pure system the density operator is given by: ρˆ = |ψihψ| ,
(42)
and in the case of a mixed state by: ρˆ =
X i
pi |ψi ihψi | .
(43)
As before, pi is the probability to find the system in state |ψi i. Now, we know, that in the case of a pure system we have: ρˆ2 = |ψihψ|ψihψ| = |ψihψ| = ρˆ ,
(44)
and therefore: ρ) = 1 . Tr ρˆ2 = Tr (ˆ
In the case of a mixed state we find: ρˆ2 =
X ij
pi pj |ψi ihψi |ψj ihψj | =
X i
(45)
p2i |ψi ihψi | .
(46)
This means: X X X 2 p2i . Tr ρˆ2 = pi hk|ψi ihψi |ki = p2i hψi |kihk|ψi i = ik
ik
i
(47)
6 P We also know that i pi = 1. However, p2i cannot be greater then pi : 0 ≤ pi ≤ 1. This means that we have p2i < pi , and therefore, in the case of a mixed state: (48) Tr ρˆ2 < 1 . Let us investigate a simple example: one spin with S = 1/2. In this case the density operator ρˆ is given by: ρˆ =
1 ˆ ˆ ·σ ˆ . 1 + hσi 2
(49)
ˆ is the identity matrix. Then, we have ˆ = (ˆ σ σx , σ ˆy , σ ˆz ) are the Pauli matrices and 1 Tr ρˆ2
2 1 ˆ + hσi ˆ ·σ ˆ Tr 1 4 1 ˆ2 ˆ · σ) ˆ 2 + hσi ˆ ·σ ˆ . = Tr 1 + (hσi 4 =
ˆ 2 = 1, ˆ Tr(σ) ˆ [see Eq. (5)]. Therefore, we get: ˆ = 0, and Tr(ˆ We know, 1 σα2 ) = 1 1 + hˆ σx i2 + hˆ σy i2 + hˆ σz i2 ˆ 1 . Tr ρˆ2 = Tr 4
(50)
(51)
ˆ = 2 in the case of the 2 × 2 identity matrix 1 ˆ we get finally: With Tr(1)
1 + |hσi| ˆ 2 ≤1, (52) Tr ρˆ2 = 2 Tr ρˆ2 = 1 in the case of a pure system and Tr ρˆ2 < 1 in the case of a mixed state. ˆ = 1 and for a mixed state |hσi| ˆ < 1. And, this means that hσi ˆ has a This means we find for a pure state |hσi| ˆ depends on the mixture and can conserved length if the system is in a pure state. In the case of a mixed state |hσi| vary with the time. VI.
VON NEUMANN ENTROPY
A good way to quantify the entanglement of a quantum mechanical system is to use the von Neumann entropy: S (ˆ ρA ) = −Tr (ˆ ρA log2 ρˆA ) .
(53)
The von Neumann entropy can be seen as the quantum mechanical analog of the classical Shannon entropy known from the information theory [4]. ρˆA is the reduced density operator defined before in section IV. If the system shows no entanglement the von Neumann entropy is zero: S (ˆ ρA ) = 0. However, a nonzero value: S (ˆ ρA ) > 0 means entanglement. The value of S (ˆ ρA ) increase with increasing entanglement. Alternative to Eq. (53), we can calculate the von Neumann entropy via: X λi log2 λi , (54) S (ˆ ρA ) = − i
ˆ = 0. where λi are the eigenvalues of ρˆA : det ρˆA − λ1 Please notice: in the case λi = 0 the von Neumann entropy S (ˆ ρA ) becomes: lim S (ˆ ρA ) = 0 .
λi →0
(55)
To get a better understanding the following example shall be discussed: A quantum system with three spins with spin quantum number S = 1/2. The wave function in this case is given by: X (56) cm1 m2 m3 (|m1 i ⊗ |m2 i ⊗ |m3 i) . |ψi = m1 m2 m3
The basis states |mi i = |Si = 12 , mi i, i ∈ {1, 2, 3}, can be | ↑i or | ↓i, corresponding to a spin oriented up or down with respect to the quantization axis.
7 Then the density operator ρˆ of the complete system is given by: X X ρˆ = |ψihψ| = cm1 m2 m3 c∗m′1 m′2 m′3 (|m1 ihm′1 | ⊗ |m2 ihm′2 | ⊗ |m3 ihm′3 |) .
(57)
Or written in matrix form: |c↑↑↑ |2 c↑↑↓ c∗↑↑↑ c↑↓↑ c∗↑↑↑ c↑↓↓ c∗↑↑↑ ρˆ = c↓↑↑ c∗↑↑↑ c↓↑↓ c∗ ↑↑↑ c↓↓↑ c∗ ↑↑↑ c↓↓↓ c∗↑↑↑
(58)
m1 m2 m3 m′1 m′2 m′3
c↑↑↑ c∗↑↑↓ |c↑↑↓ |2 c↑↓↑ c∗↑↑↓ c↑↓↓ c∗↑↑↓ c↓↑↑ c∗↑↑↓ c↓↑↓ c∗↑↑↓ c↓↓↑ c∗↑↑↓ c↓↓↓ c∗↑↑↓
c↑↑↑ c∗↑↓↑ c↑↑↓ c∗↑↓↑ |c↑↓↑ |2 c↑↓↓ c∗↑↓↑ c↓↑↑ c∗↑↓↑ c↓↑↓ c∗↑↓↑ c↓↓↑ c∗↑↓↑ c↓↓↓ c∗↑↓↑
c↑↑↑ c∗↑↓↓ c↑↑↓ c∗↑↓↓ c↑↓↑ c∗↑↓↓ |c↑↓↓ |2 c↓↑↑ c∗↑↓↓ c↓↑↓ c∗↑↓↓ c↓↓↑ c∗↑↓↓ c↓↓↓ c∗↑↓↓
c↑↑↑ c∗↓↑↑ c↑↑↓ c∗↓↑↑ c↑↓↑ c∗↓↑↑ c↑↓↓ c∗↓↑↑ |c↓↑↑ |2 c↓↑↓ c∗↓↑↑ c↓↓↑ c∗↓↑↑ c↓↓↓ c∗↓↑↑
c↑↑↑ c∗↓↑↓ c↑↑↓ c∗↓↑↓ c↑↓↑ c∗↓↑↓ c↑↓↓ c∗↓↑↓ c↓↑↑ c∗↓↑↓ |c↓↑↓ |2 c↓↓↑ c∗↓↑↓ c↓↓↓ c∗↓↑↓
c↑↑↑ c∗↓↓↑ c↑↑↓ c∗↓↓↑ c↑↓↑ c∗↓↓↑ c↑↓↓ c∗↓↓↑ c↓↑↑ c∗↓↓↑ c↓↑↓ c∗↓↓↑ |c↓↓↑ |2 c↓↓↓ c∗↓↓↑
c↑↑↑ c∗↓↓↓ c↑↑↓ c∗↓↓↓ c↑↓↑ c∗↓↓↓ c↑↓↓ c∗↓↓↓ c↓↑↑ c∗↓↓↓ c↓↑↓ c∗↓↓↓ c↓↓↑ c∗↓↓↓ |c↓↓↓ |2
.
Only the red matrix elements contribute to the reduced density operator ρˆA = ρˆm1 , corresponding to the first spin: ρ) . ρˆm1 = Trm2 m3 (ˆ
(59)
Trm2 m3 is the partial trace over the contributions coming from the second and third spin. All the information, about the system, stored in the black matrix elements get lost during the calculation of the reduced density operator ρˆm1 . Then, we can write: X (hm′′3 | ⊗ hm′′2 |) ρˆ (|m′′2 i ⊗ |m′′3 i) ρˆm1 = ′′ m′′ 2 m3
=
X
X
m1 m2 m3 m′1 m′2 m′3
=
X X
m1 m′1 m2 m3
X
′′ m′′ 2 m3
cm1 m2 m3 c∗m′1 m′2 m′3 (|m1 ihm′1 | ⊗ hm′′2 |m2 ihm′2 |m′′2 i ⊗ hm′′3 |m3 ihm′3 |m′′3 i)
cm1 m2 m3 c∗m′1 m2 m3 |m1 ihm′1 | .
(60)
All sums are over the possible configurations ↑ and ↓. Then we can write the reduced density operator ρˆm1 in matrix form as: c↑↑↑ c∗↓↑↑ + c↑↑↓ c∗↓↑↓ + c↑↓↑ c∗↓↓↑ + c↑↓↓ c∗↓↓↓ |c|2↑↑↑ + |c|2↑↑↓ + |c|2↑↓↑ + |c|2↑↓↓ ρˆm1 = . (61) c↓↑↑ c∗↑↑↑ + c↓↑↓ c∗↑↑↓ + c↓↓↑ c∗↑↓↑ + c↓↓↓ c∗↑↓↓ |c|2↓↑↑ + |c|2↓↑↓ + |c|2↓↓↑ + |c|2↓↓↓ Due to the fact that for S = 1/2 we have only two configurations up and down the reduced matrix becomes 2 × 2. In the case S = 1 the matrix would be 3 × 3 and so on. ˆ = 0. As result Then, the eigenvalues of this matrix are easily calculated using the standard method: det ρˆA − λ1 we find: r α+δ (α + δ)2 ± − (αδ − βγ) , (62) λ1,2 = 2 4 with α β γ
= = = and δ =
|c|2↑↑↑ + |c|2↑↑↓ + |c|2↑↓↑ + |c|2↑↓↓ , c↑↑↑ c∗↓↑↑ + c↑↑↓ c∗↓↑↓ + c↑↓↑ c∗↓↓↑ + c↑↓↓ c∗↓↓↓ , c↓↑↑ c∗↑↑↑ + c↓↑↓ c∗↑↑↓ + c↓↓↑ c∗↑↓↑ + c↓↓↓ c∗↑↓↓ , |c|2↓↑↑ + |c|2↓↑↓ + |c|2↓↓↑ + |c|2↓↓↓ .
Now, we know |ψ(t)i as function of time t. Therefore, we also know cm1 m2 m3 (t), ρˆ(t) and ρˆm1 (t), and with (62), we get the eigenvalues λi (t) which we need to calculate the von Neumann entropy S(ˆ ρm1 (t)). The information loss during reducing the density operator from ρˆ to ρˆm1 can be seen in the following example: three spins with S = 1/2 in the AFM ground state.
8 In this case the wave function is given by: 1 |ψi = √ (| ↑↓↑i − | ↓↑↓i) , 2
(63)
or in binary code:
1 |ψi = √ 2
0 0 1 0 − 0 0 0 0
0 0 0 0 1 0 1 0 0 . = √ 0 0 2 1 −1 0 0 0 0
Then, the density operator in matrix form is given by: 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 ρˆ = 0 0 1 0 0 −1 0 0 = 2 0 20 0 0 0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
(64)
0 0 −1 0 0 1 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
.
(65)
Here, the same color code has been used as before: the red numbers contribute to the reduced density operator ρˆm1 . The information stored in the black matrix elements get lost. The result is: 1 1 0 . (66) ρˆm1 = 2 0 1 ρˆm1 is the reduced density matrix of a single spin (in this case the first spin) in pure spin state with three spins S = 1/2. The important point is, ρˆm1 is identical to the density operator of a mixture with 50% up spins and 50% down spins. The eigenvalues of this matrix are λ1 = λ2 = 1/2, and therefore the von Neumann entropy S (ˆ ρm 1 ) = 2(−0.5log2 0.5) = 1, which means that this state is highly entangled.
[1] [2] [3] [4] [5]
Ugo Fano, Rev. Mod. Phys 29, 74 (1957). Valery L. Pokrovsky and Nikolai A. Sinitsyn, Phys. Rev. B 67, 144303 (2003). Paul A. M. Dirac, Math. Proc. Cambridge Philos. Soc. 26, 376 (1930). Claude E. Shannon, The Bell System Technical Journal 27, 379 & 623 (1948). U = (v1 , . . . , vN ) is a row vector which contains as columns N the eigenvectors vi of the density operator ρˆA as column vectors.
